Calculus
6th Edition
ISBN: 9781465208880
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 13, Problem 79SP
To determine
To evaluate: The given line integral.
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Evaluate the line integral
(3ry² + 6y) dr, where C is the path traced by first moving from the
point (-3, 1) to the point (2, 1) along a straight line, then moving from the point (2, 1) to the
point (5,2) along the parabola x = y² + 1.
Let C be the square with vertices (0, 0), (1, 0), (1, 1), and (0, 1) (oriented counter-clockwise).
Compute the line integral: y² dx + x² dy two ways. First, compute the integral directly
by parameterizing each side of the square. Then, compute the answer again using Green's
Theorem.
Calculate the line integral
f (10xy² + 9x + 3y − 5) dx + (7xy − 8) dy,
where C' is the rectangle with vertices (1, -3), (1,0), (-2, 0), and (-2,-3) oriented counterclockwise. Enter an exact
answer.
Provide your answer below:
(10xy² +9x+3y-5) dx + (7xy − 8)dy =
Chapter 13 Solutions
Calculus
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- Calculate the line integral f (−x²y − 3x + 6y + 7) dx + (7xy + 1) dy, where C' is the rectangle with vertices (−2, −3), (−2, 2), (-4, 2), and (-4,-3) oriented counterclockwise. Enter an exact answer. Provide your answer below: fc(-x²y-3x+6y + 7) dx + (7xy+1) dy =arrow_forwardLet the curve C be the line segment from (0, 0) to (3, 1). Let F = ⟨2x-y, 4y-x⟩ Calculate the integral ∫c F· dr = ∫c (2x-y)dx + (4y-x)dy in two different ways:(a) Parameterize the curve C and compute the integral directly. (b) Use the Fundamental Theorem of Line Integrals.arrow_forwardShow that the function given by f (x, y) = x² + 3xy is differentiable at every point in the plane. By using the definition.arrow_forward
- Suppose F(x, y) = x² + y² and C is the line segment from point A = (1, −1) to B = (3, −5). (a) Find a vector parametric equation (t) for the line segment C so that points A and B correspond to t= 0 and t = 1, respectively. F(t) = = (b) Using the parametrization in part (a), the line integral of along C' is dt LE · dF = [ ° F (F(t)) - 7"' (t) dt = √° with limits of integration a = and b = (c) Evaluate the line integral in part (b).arrow_forwardConsider the line integral xy dx + x²y° dy with C the triangle with vertices (0, 0). (1,0), and (1, 2). Evaluate this line integral by two methods: (a) directly, and (b) by using Green's Theorem. Orient C in the counterclockwise direction. (,?) (1,0) Hint: the curve C'is composed of three line segments, so you will have to compute the line integral along each segment separately.arrow_forward4) Evaluate the integral (2e-2x) dy + (2xy +1) dx using Green's theorem where C is C 64 consists of y =4-x² and straight line from (-2,0) to (2,0). Sketch the curve C. ans: 3arrow_forward
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